— ? —__

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I. RIPO~~? NUMBER 2. OOV1~ ACCEUION 140 3. RECIPIENT~5 CATALOG NuM•ER

Technical Report No. 135 -

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UCTION PLANNING~~

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S. PERFORMING ORGANIZATION NAM E AND ADDRESS t O. PROGRAM ELEMENT. PROJECT TASKAREA S BORIC UNIT NUMBERS

M . I . T . Operations Research Center77 Massachusetts Avenue NR 347—027Cambridge , MA 02139

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14. MONITORING AGENCY NAME a AOORESS(II dSIt.,w t from ContvollMj OUlc.) IS. SECURITY CLAS1~7~I

ISa. OECL ASSI FICATION/OOWNGRADINGSCHEDULE

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Releasable without limitation on disseminat ion .

17. DISTRIBUTION STATEMENT (of A. .b.lracE i1.r.d hi Block 20, II dltI.r nl fro. R.port) -

IS. SUPPLEMENTARY NOTES

~2~~2 7~15. KEY WORDS (Conlhw. on to,.,.. .Id. if n.c..o.y aid id.neity by block nia,b.r)

Hierarchical Planning SystemsDemand Forecasting

20. A BSTRACT (Cordinu. on r.va~ .l~~~ if , . c o o y aid Id.aIi ~~ b~ block no.k.c)

See page 1.

DO , ~~~~~~~ 1473 EDITION O~ MOV U15 OBSOLETE ZJncl as si f i ed 4~7

SECURITY CLASSIFICAT ION OF rrns PAGE (~~~s, DMa tniar.d)

HIERARCHICAL PRODUCTION PLANNING SYSTEMS

by

ARNOLDO C. MAX

and

JONATHAN J. GOLOVIN

Technical Report No. 135

Work Performed Under

Contract N000 14—7 5—C—0556 , Office of Naval Research

Multilevel Logistics Organization Models

NR 347—027 M.I .T. OSP 82491

Operations Research Center

Massachusetts Institute of Technology

cambridge, Massachusetts 02139

Section ~jo

August 1977

Y)J~A~J1T ~OF~SPECIAL

‘-1-.

Abatraa t.~ir This describes the development of hierarchicalplanning systems to support nedii.mi range planning and operationaldecisions in a batch processing production environment . In thisapproach , higher level decisions impose constraints to lower levelact ions, arid lower level decisions provide the necessary feedbackto reevaluate higher level actions . An analysis of the existingmet~~dolo~ r to design hierarchical production systems is given.Computational results are presented.

1. Introduot~on

Production can be defined as the process of converting raw iraterials

into finished products. An effective managerent of the production process

sheuld provide the finished products in appropriate quantities , at the

desired tiz~ s, of the required quality, and at a reasonable cost.Production management encompasses a large nt.nnber of decisions that

affect several organizational echelons • These decisions can be ~~~uped

into three broad categories: (1) strategic decisions, irwolving policy

tornu~lation, capital investment decisions, and design of physical f~cili—

ties; (2) tactical decisions, dealing prizrerily with a~~ ’egete production

planning; and (3) operational decisions, concerning detailed production

scheduling issues. !I~~se three categories of decisions differ merkedly in

tent~ of level of nenagement responsibility and interaction, scope of the

decision, level of detail of the required infon~ation, length of theplanning I~ rizon r~eded to assess the consequences of each decision, andde~~ee of uncertainties and risks inherent ira each decision. ~~~se canal—

derations have led us to favor a hierarchical planning system

to support production nenagement decisions, which guarantees an appropriate

coordination of the overall decision making process , but , at the sai~ time,

recogoizes the intrinsic characteristics of each decision level . A

—2—

justification for this hierarchical approach and its inplications for the

design of a production system has been reported by Hax [U.). Early rrcti-

vation for this approach can be found in the pioneering work of Holt ,

1’bdigliani, Muth, and Siircri [15], and in Winters [20]. Hax [3 1 described

an application of a hierarchical production system for a continuous manu-

facturing process. Hax and ~~al 1114], and Bitran and Hax [1) addressed

the use of hiearchical systems in a batch processing envirornnent . Annstrong

and Hax [3], and Shwiir~r [19] analyzed an application for a job shop activity .

I~ cent theoretical research in the field of hierarchical production planning

systems has been conducted by Golovin 110), (]abbay [ 9) , and Candea [61.

This chapter discusses the general issues associated with the design

of hierarchical production planning systems. An overall description of

the characteristics of such systems is given in SectIon 2. SectIon 3 ana-

lyzes t;he ag~ ’e~~te production planning decisions. Section 4 justifies the

need for hierarchical planning systems. Section 5 presents the treatrr~nt -

of demand forecasts. SectIons 6 and 7 discuss the nx st important methodo-

logies proposed to dlsa~~r’e~~te hi~~~r level decisions. 1’inally , Section 8

provides oonputatlonal results corparing the efficiency of the various

d.taa~~ e~~tlon metho~~logles. -

2. A Hiezv.rchioai Production Plan ning System

Production decisions involve complex choices an~ng a large number of

alternatives. ~~~se choices have to be made by trading-off conflicting

objectives under the presence of financial, techr~ logical, and marketing

constraints • Such decisions are mt trivial and nodel based systems have

proven to be of ~ ‘eat assistance in supporting mar~gerial actions In this

field . In tact, one could argue that , in this respect , production is the

nost nature field of management. A ~ ‘eat many contrIbutIons have been made

in this field by operations research, system analysis, and computer sciences .But now we believe it Is both sIgnifIcant and feasIble to attempt a rrore

cariprehenslve and inte~ ’ative approach to productIon management.

1~~ optimal planning and scheduling of nialtlple products has received

nvch attention in the operations research literature. Several attempts

(Marine (17], Dzielinski, ~aker , and Marine [7], Dzielinski and Gonory [8],

Lasdon and Terjung [16]) have been made to formulate the overall problem

as a single mixed—integer mathematical pro~~anining nodel to be solved on

a rolling horizon basis . However, these approaches require data such as

the forecast demand for every item over a complete seasonal cycle, usually

a full year. When these systems involve the scheduling of several thousands

of it~~s, these data requirements become overwhelming and the resulting

planning process becomes unrealistic due to the magnitude of the forecast

errors inherent in such detailed long term forecasts.

~~ obvious alternative to a detailed nonollthic approach to produc-

tion planning Is a hierarchical approach. ‘~L~~ basic design questions of

a hierarch.IcaJ. planning system are the partitioning of the overall planning

problem and the linkage of the resulting subprob].ems. An Important input

to resolve these questions is the rrznber of levels recognized in the product

structure • Hax and 1~~a1 j 1J 4 J identified three different levels :

(I) Items are the final products tL be delivered to the customers. They

represent the hig~ st de~ ’ee of specificity re~~rding the manufactured

products. A given product may generate a large ntziter of items differi ng

In ten~ of characteristics such as color, packaging, labels, accessories ,

size, etc.

(ii) Families are gnxips of items which share a conron manufacturing

setup - cost • Economies of scale are accomplished by join tly replenishing

•1~-

~~-

Items belonging to the same family.

(iii ) Types are groups of families whose production quantities are to bedetermined by an aggregate production plan. Families belonging to a type

normally have similar costs per unit of production time, and similar

seasonal demand patterns .

We have found that these three levels are necessary to characterize

the product structure in many batch processing manufacturing environments

we have examined. Obviously, there are some practical applIcations where

there are some practical applications where additional or fewer levels mightbe needed. In the remainder of this paper we will propose a hierarchicalplanning system based on these three levels of item aggregatIon. Note,however, that conceptually the system can be extended to any number ofa~~ egation levels by defining appropriate subproblems linking those levels.

‘fl~ first step in our hierarchical planning approach is to allocatethe total production capacIty anong product types by means of an aggregateplanning nodel. The planning horizon of this nodel covers rorsally a ful lyear in order to properly consider the fluctuating demand requirements forthe products. We advocate the use of a linear prograxrgning nodel at thislevel. ¶fl-~re are various advantages associated with the use of such anodel that will be addressed in the next section. The major drawback isthat a lix~ ar prograzTxnin,g nodel does rot take setup costs into consideration.‘Itt isTpllcatlons of this limitation will be examined in detail in SectIon 7,

The second step in the planning process is to allocate the productionque.ntities for each product type anong the families belonging to that type.This Is done by disa~~ ’egating the results of the a~~regate planning nodelbut only for the first period of the planning horizon, thus substantiallyreducing the required azix unt of data collection and data processing. Thed.tsa~~regetIon procedure used assures consistency and feasibility anong

-5-

the type and family production decisions while attempting to minimize the

total setup costs incurred in the production of families • It is at this

stage where setup costs are row explicitly considered .

Finally, the family production allocat ion is divided anx ng the items

belonging to each family . The objective of this decision is to maintain

all Items at an inventory level that maximizes the time between family

setups. Again, consistency and feasibility are the driving constraints

of this disa~~ ’egat ion process.

An extensive justification for this approach is provided In Hax (11]

and Hax and ?~ al [114]. Under certain cost structures, it has been shown

to be optimal (Gabbay [9]) . Under nore general cost structures, It has

empirically been found to perform exceptionally well, as discussed in

Section 7. FIgure 1 shows the overall conceptualization of the hierarchi—

Tiead In last period’s usa~e

Update 1r-.v~r~tcry ~tatus( Physical Inventory , Arr~unt on

Order , Backorders,Available Ir:,entory)

Update demand forecas;s , safetystocks, overstock limits , and

I-—— I’Ufl OUt times4

Determine effective derands[

~ for each product type

.J Aggregate Plan for Types‘L(Aggregate Planning Reports)

4. _ _j Family Disaggregation J r~agemerv~A~~ (Family Plarnlng Reports) ~ )( InteractI~~

I Item Disaggregat iont (Item P1annl r.~ Pecorts)

4.L Detailed Status Report s

FIGU~~ 1 • ( EP’IIJAL OVERVI~~ OF }~~RA~~}~ CAL PLANNING SYS~~4

-6-

ca]. planning effort. A computer based system has been developed to faci-

litate its Irnplerientation . The details of this system are reported in

the next chapter of this book (Hax and Golovin [ .3 ] ) . Herein we will

concentrate on the mathodological issues associated with the system design.

3. Aggrega te Production PZ.anning for Typ es

This Is the hI~~est level of planning in the production system, addressed

at the type level. Essentially any aggregate production planning model

can be used as long as it adequately represents the practical problem under

consideration. (For extensive discussions of possible aggregate models,see &iffa and ~~ubert [5] , Hax 1121, arid SIlver [18].) We consider the

following sirrplified linear program at this level:

I T TMinimize E E (cjtXjt + hitIjt)+ I (r Rt + o 0 )

i—l t=]. t—]. t t

subject to:Xft — ‘i,t+L + ‘i t+L—l — dj,t+L , 1],...,I; t—1,...,T

I

~~ miX1~ ~~

. 0~ + R~ ,

Rt ~~~. ~

‘~~t , t—l,...,T

0t ~ (oin)~ , t 1,...,T

x1~ > 0 , i—l,...,I, t—l ,...,T

14 4. > 0 , i—l,...,I, t L+l,...,L4’I

Rt, 0t > 0 , t—l ,...,T

~~~~~~~ decision variables of the nodel are : X1~, the n~znber of units to be

produced of type I during t; I~ ~~~~~~~~~

the n~.rte r of unite of inventory retain-

—7—

Ing at the end of period t+L; Rt and at’ the regular and overtime hours

used during period t , respectively.

~Ihe parameters of the model are: T, the length of the planning horizon;

L, the length of the production lead time; cit, the unit production cost

(excluding labor); hit, the inventory carrying cost per unit , per period;

rt and the cost per manhour of regular and overtime labor, respectively;

(rm )t and (om)~,

the total availability of regular and overtime hours in

period t, respectively; and mi, the inverse of the productivity rate for

type 1, in hours/unit. di,t÷L is the effective demand for type I during

period t+L. (For a definition of effective demand, see section 5.)

Whenever the production costs cit are invariant with time, no back—

orders are allowed, and the regular work force payroll is a fixed corrrnitment ,I T T

the terms I E c4 X,, ~.

and I rtR~ are fixed and should be deleted from11 t—l t—l

the objective function. In that case, the model seeks the optisruxn aggre-

gate plan trading off inventory holding and overtime costs. It is straight—

for~~rd to extend the model to include other cost factors and decisions,

such as hiring and firing, backorders, subcontracting, lost sales, etc.

Also, the constraints can represent art’ number of technological, financial,

marketing restrictions, or other considerations.

Linear prograxnidng is a convenient model to use at this aggregate

level due to both its computational efficiency and the wide availability

of linear programning codes . In addition, L.P. permits sensitivity and

parametric analyses to be per formed quite easily. The shadow price infor—

niation that becanes available when solving L.P . models can be of assistance

In identifying opportunities for capacity expansions, market penetrations ,

Introduction of new products, etc.

Notice that the manufacturing setup costs have been purposely ignored

in this aggregate model forn ~ilat Ion. In practice , we have found that setup

• -8-

costs have a secondary impact on the total production cost (see Section 8

for sensitivity analysis). Moreover, the inclusion of setup costs would

force the model to be defined at a family level. This implies a high

level of detail which invalidates all the advantages of hierarchical

planning discussed in the next section. Consequently, setup costs are

considered only at the second level of the hierarchical planning process.

Because of the uncertainties present in the planning process, only

the first time period ’s production plan is implemented. At the end of

each time period, new infonrat ion becomes available that is used to update• the model with a rolling planning horizon of length T. Therefore , the

data transmitted from the type to the family level is the resulting

product typce production and inventory quantities for the first period

of the a~~ ’egate model . These quantities will then be disaggregated

among the famil ies belonging to each corresponding type.

~~. The Need for Hierarchical Planning

1~~ advantages of the aggregate approach as compared to a detailed

one nay now be clearer . These advantages can be divided into three

distinct categories.

‘ft~ first category considers the costs of data col).ection to support

the model as well as the computational cost of running the model. A

major infoni~t ion system may be required to collect the demand productivity

and cost data as well as prepare forecasts for thousands of individual

items, a more costly project than building the production planning system

itself. This data must then be reviewed by management . As the number

of items Increases, this effort can become unwieldy , leading to deter lora—

-9-

tion of the data used in the planning process and therefore the output .

In most cases, this cost of data collection and preparation will far out-

weigh the cost of computation. This is import ant to note as the cost of• computation continues to decrease and it becomes feasible to solve enonrous

linear or non—linear programming problems. Aggregation of items can• sigoificantly reduce the cost and effort in demand forecasting and data

• preparation in addition to decreasing the computational costs.• ¶fl~ second categery considers the accuracy of the data . Unless all

items are perfectly correlated , an aggregate forecast of demand will have

reduced variance. In general, we are able to employ more sophisticated

techniques such as econometric models or auto regressive-moving average

statist ical models and spend more tiii~ in obtaining managerial j ud~~ent ,

given the smaller number of forecasts required. Since decisions on regular

time, overti.j r~ , hiring and firing , and other production rate changes are

based on the total production quantity demanded, increased forecast accuracy

on total demand should improve the decision making process.

Finally, and perhaps from an implementation standpoint, trost iiTIpor-

tantly, a~~ ’egation leads to more effectI’ie managerial understanding of

the model’s results. When ten thousand items are being planned simulta-

neously, the sensitivity of the results to changes in individual item denrids

may be complex. There are too many combinations of changes to consider.¶L~~ manager may never be able to see the overall picture but, instead, be

lost in the details.

In addition, at this level of managerial planning, most marketing

forecasts are made by product group and decisions made by product line or

manpower class. These are budgeting decisions, not lot sizing decisions

for next week. It is crucial that the decision variables and sensitivity

- • arml ~ sis that can be carried out correspond to those with which the manager

• —• -~~~~~~~~ •• - - •s•• - - • ~~ •— —

—10—

deals.

While all these argu ment s for a~~ ’egation are valid , they would be

meaningless if it were not possible to ciisag~ ’egate back to the detailed

level and obtain near optimal results from this hierarchical approach.

Our results , discussed in Section 8 have shown hierarchical systems to be

near optimal under a variety of realistic conditions.

5. Demand Fore casts

Unless care is taken , the use of aggregation may lead to infeasibili—

ties. It is ijr~r~”~tant to realize that inventories and demand only have

physical meaning at the item level. The concept of product types is a

mere abstraction that makes possible the aggregation process. When

calculating product type inventories, it is incorrect to sinip].y add the

inventorie s,of all the items belonging to a product type. Implicitly,

that pr actice assi.ui~s complete Interchangeability of the inventories an~ng

all the items in a product type, which is not the case . To illustrat e this

• point, consider a product type consisting of items 1 and 2, whose initialInventories and demand requirements for the next five period s are as follows :

I~ mand by PeriodInitial Inventory 2

_~~~_

Item 1 600 100 100 200 200 1~00Item 2 100 200 200 kOO ~O0 800

Total 700 300 300 600 600 1200

By simply considering total product type demand and inventory, we ~~uld

calculate net demands of 0, 0 , 500 , 600 , and 1200. But, in fact , we will

run Out of Item 2 in perIods 1 and •2. The problem arose from assuming

• -11-

that we could use product type inventory held in item 1 for Item 2. ThIs

problem is corrected by defining effective demands for each item.

Formally, ~~ is the forecast demand for item k in period t, AIkIs its corresponding available inventory, and

~~k is Its safety stock, the

effective demand dk t of item i for period t is given by:

= max dk,& — AIk + ssk) , t=l,2,...,t’

(1)

d~,t — ~ c,t , t t +1,...,T

*where t is the first time period in which the Initial Inventory is depleted ,

i.e.

t’—].• E d~ — Al1, + < 0 , and

,~~~

*

- ~~1 ~~,L _ A I k + SSk > 0

~~ effective demand for a type I is simply given by the sum of the effec-

tive demands for an items belonging to a given type, i.e.

dj t = E dk (2)kcK(j) ,~~~

wtexe K(I) is the set of all items k belonging to product type I.

In our previous example , the effective demands are:

Effective 1~ mand by Period

Item i 0 0 0 0 1400Item~~ 100 200 1400 1400 800

Total Effective t~mand tbr 100 200 14 14Product ~rype 00 00 1200

The hierarchical forecasting system operat ~s as follows:

(I) An ag~ ’e~~te forecast is ~~r~rated for each pro~~ct type for each

-12—

time period In the planning horizon.

(U) The type forecasts are disa~~ ’e~~tec! down to Item fo~’ecasts. This

disa~~’e~~tion can be done by forecasting the proport ion of the total type

demand corresponding to each item. These proportions can be updated using

exponential sn~othing techniques, which are appropriate to apply for a

short horizon at a detailed level . Notice that item and family forecasts

are only required for a few time periods in the product type disa~~ ’egation

n~de1s we will present .

(iii ) After updating the available inventory for each item, the effective

item demand is calculated by applying expression (1) above . Whenever the

initial available inventory exceeds the first period’s demand, expression

(1) requires item forecasts for successive periods in the planning horizon.‘lt~se forecasts can be obtained by making trend and/or seasonability adjust-

ments to the Initial period forecasts , age.In using exponential sn~othIng

techniques.

(iv) The effective demand for types is obtained ft’czn expressIon (2).

These demands are used in the aggre~~te n~de1 described in section 3.O~rputer programs to perform automatically the necessary calculations axe

discussed in the next chapter.

Note that this forecast ing system is an example of a top down approach,using a~~’e~~te product type forecasts and disa~~re~~ting, rather than

using detailed forecasts and s~iining to ~~t the a~~’e~~te product type

forecasts.

6. The Poinii~j D~aaggr.ga tion Moda l

The central condition to be satisfied at this planning level fbi’ a

—13-.

coherent disaggregation is that the si.un of the productions of the families

in a product type equals the an~unt dictated by the higher level n~del

(plan ) for th is type . This will assure consistency between the aggregate

production plan and the family disaggregation process. This consistency

should be achieved while determining the run quantities for each family

that minimize the total setup cost arrong families, the remaining cost to

be considered.

~~ will row examine four disaggregation methods which have been pro-

posed in the literature : Hax and Meal ( lL4 ] , Knapsack [1], Winters [20],

and Equalization of I~n Out Times.

6.1 Max and Meal Method

Conceptually, Max and Meal [1k ] suggested a heuristic approach to

family disaggregation that:

(I) schedules those families in each product type that rrn~st be run in

the current planning period in order to meet the items’ service require—

merits;

(ii) sets initial family run quantities so as to minimize cycle inventory

and setup costs. (This is accomplished by setting the Initial family run

quantity equal to the corresponding family economic order quantity.);

(iii ) then adjusts the run quantitites of the families so as to use all

the production time allocated to each product type by the ag~ egate planning

nodel, while observing items’ overstock limits.

To liiplement these three conditions, Max and P~al proposed an algoritbn

with the following rules:

(I) Only those families which trigger during the current planning period

have to be scheduled ror production. A family is said to tri~~er whenever

-1L~-

the current available inventory of any of its Items carrot absorb itsexpected demand during its productio n lead time plus the review period,i.e. whenever

AI.~ < ~dk,l + dk,2 + ... + dk,L+1) + SSk for an~r kcK (J ) ,

where K(j ) is the set of all items k belonging to family j.Equivalently , we can define the run out time of an item k to be

AIk~~~SSFOT kk L+l

E d kt~1A family j is said to trigger whenever its run out time for at least oneitem is less than one time period, i.e.

I~T = Mm FIYT = M.tn ~~~ -

~~~ < 1kcK(j ) k keK(j) L4l

t—l ‘1C,t

If we do not produce this family j some mem ber item will run out , violating

our ass~ iptIon In the aggregate plan of no backorders.

(ii ) ~~ initial run quantity, Y~, for a family that has triggered, Is

set to the miniimzn between its economic order quantity, EOQ~; and the

difference between the overstock limit of the family, OS~, and Its current

available inventory, AI~; I.e.

— Min (EOQ~~ OS~~_ A I ~)

where

AZ — E AI.~kcK(j )

OS — £ OS~ , andkcK(j )

—15—

EOQ~ can be determined by the lot size formu la for a family of related

items (&‘own [‘~ ,, page i47) . When an item k has a terminal denend at the

end of a season , OSk can be calculated by means of a newsboy nodel (see

ZixTIrere~m and Soverei~ i (21 1, page 370).

( iii ) If the sum of the initial run quantities for the families belonging

to a product type I does rot add up to the total production time allocated

to that type by the aggregate plan ning ~odel, adj ustments in the family

run qua ntitie s are needed.

Let J(i) be the set of all families belonging to product type I, and

be the total production to be allocated arrong these families . has been

determined by the aggregate planning nI)del, and corresponds to the optimal

value of va~iab1e Xil since only the firs t period result of the aggregate

ziodel is to be i~~lenented. Two cases should be considered :

(a) If 1 Y < X~, the new run quantities for each families are :jcJ(i) ‘~

a kcK(j )Y — Mm I E (OSk~AIk), Y + (X1— E Y~) — -

~ (OS~ —Al.)j [kcK(j ) jcJ(i) jcJ(i) kcK(j ) K K

This simply states that the initial, run quantities are expanded In propor-.

tion to the difference between the overstock limits and the current avail-

able inventory. This difference corresponds to the mexlmurn allowable produc-

tion of each family, up to its overstock limit. If all the families that

have tri~~~red must be produced to their overstock limit , and the total*production is still lees than the aggregate production requirement X1, one

should ~~ deeper in the run out list scheduling families belonging to

product type i In order of increasing run out time until we reach the total

aseI~ ied production capacity . ‘I~~ new famil ies should be run ~~ to their(

—16—

naxirr ~ n allowable quantities.

(b) If E Y 4 > X~, the run quantities are decreased In proportion toj cJ(i)

their initial assi~ irents , i.e.

* * YjYj — X~ ~j~J(i) ~

For further details about this methodolo~ r , the reader is referred

to []M] .

6.2 X.nap8ack Method

Bitran and Hax (1] prop osed a disaggregation technique which essentially

is a fonmllzation of the heuristics developed by Hax and ~~al. For

every product type I the following convex Ia~apsack problem is solved:

s4 dMinimize E

jcJ(I) ~j

5ubject to: *£ y

jcJ(i) ~

tb3 < Y1 < u b 3, jcJ(i) (3)

where is the nunter of unit 8 to be produced of family j , is the setup

cost for family 3, d3 Is the forecast denar~I for family 3 (usually anazmial. forecast de~~nd), Lb~ and ub3 are lower and upper bounds for the

qantity Y1, and Is the total anount to be allocated an~~g all the

families b elonging to type i.

¶1~* lo~~r bound Lb3, that defines the minhintxn production quantityfor family J , is given by:

— nax [o. Cd311

+ d~~2 + ... + d31~~.1

) — Al3 + S53 }

—17—

This lower bound, Lb , the minlnuim production to avoid backorders given

current forecasts , guarantees that any backorder’s will be due only to

forecast errors greater than the safety stock S33.¶I~~ upper bound ub

3 is given by:

ub3

— OS,~ — A l3

where OS3 is the overstock limit of family 3.‘fl~ objective function assur~es that the family run quantities should

be proportional to the setup cost and annual demand for a given family.

This seems to be a reasonable assumption (and is the basis of the economic

order quantity forrrn~lation), that tends to minimize the average annual

setup cost • Notice that the total inventory carrying cost has already

been establi shed in the aggregate planning rodel; therefore it does not

enter in the current fonnulation.

The first constraint :

E —jc J( i) 3 I

*assures the equality between the a~~~’egate nodel input X1 and the sum of

the family run quantitites. It can be shown (see Bitran and Hax [11) that

this condition can be substituted by

Y < X~ (~

)j eJ(j) J I

without cheriging the optlznun solution to the disaggregation problem. Intui-

tively, the lar~~r the Y3

1 s, the mnaller the objective function value and so

the constraint Is a1~says net exactly.

Initially J( i) contains only those families which tr I~~~r during the

curre nt planning period . fle productio n for these familie s rus t be sche—

—18-.

duled In this period to avoid ft.ture backorder’s. All other families are

put on a secondary list . These families will be scheduled only if extra

capacity is available.

Bitran and Hax El ] presented an efficient algorithm to solve this

problem throu~~ a relaxation procedure. OptimaJ.ity and convergence proofs

are given in (1] and [2]. The algorithm consists in i~ x)ring initially

the boundin g constraints (3) and solving the objective function subject

to the kna psack restriction (LI). Then a check is made to verify if the

optimum values satisfy the bounds (3) . If they do, the Y,~’s constitute

the optImal solution. If not , at least soma of the Y~ ’s are shown to be

optimal and a new iterat ion takes place. The algorithm is finite because

at each iteration we determine the run quantity of at least one family.

6.3 Winters Method

Winters (20] examIned various alternatives for disaggregating the

a~~ egate production quantities. He recamended a disaggregat Ion proce-

dure in which familie3 are produced In economic order quantities, in order

of their Increasing run out tines, until the aggregate total is reached.

T~ 1ike the Max and ?~ al rrethod, the initial run quantities are not nodified,

but are treated as Indivisible discrete units and their release point is

varied.

Usir€ the tennino1o~~ described previously, the Winters disaggregation

nethod can be formalized as follows :

(I) Cc*rpite the run quantity for each family 3:

- Mm (EOQ1, OS3 — Al3

).

(ii) Coq,ute the run out tine for each family 3:

—19—

AI~~SSk= rimn kkcK(J ) dk

where d,,~ is the forecast demand for item k.

(ill) ~~nk all the families belonging to a given product type by increasi ng

run out times. This constitutes the run out list for a type.

(iv ) For each produc t type, go down the run out list accunulating Y~’suntil the total desired produ ction X1 is reac hed; i.e., produce families1 to n in the n.m out list where :

n *E Y > X 1 , and

3— 1

n-i *Z Y < X

i—i 3 1

Further discussion on this approach Is provided In (201.

6. 14 Equaiiaation of Run Out Times

An obvious alternative disaggregatlon method is to allocate the:1 production anount determined at the aggreg ate planning level for a given

type in such a vay as to equalize the run out times of all the itenebelonging to that type. This implies sld.pping the family level as a

( diaa~~~egation layer . Run out time equalizatI on is a natural disaggre-getion nethodo1o~ r to be app lied at the item level and , therefore , thecorresponding technical details will be presented in the next sect ion.

It is Important to mention at this point that when run out tine equal-Ization Is directly applied at the item level, no consideration is given

to the resulting setup costs associated with the family nu~s. Th.~s, Itmi~ it be expected that this disa~~ ’egation procedure will generate fairly

h1~~ setup costs, zelatlve to the ut.hsz’ ranLU~y Uis~~~t~~~t1Dir iit~U~~k~ L~1i~

e

—20—

take explicit account of setup costs. The possible advantages of a direct

item run out time equalizat ion are the realization of a hi~~ degree of

synchroni zation of the pro ductio n planning system , and the added simplicity

in inplementing the hierarchical system.

7. The Item Disaggregation Model

For the current planning period , all the costs have been already deter-

mined in the forner two levels and any feasible disaggregation of a family

run quantity has the same tot al cost . However , the feasible solution chosen

will establish the initial conditions for the next period and therefore

will affect future costs . In order to save setups in future periods it

seems reasonable to distribu te the family run quantity anr~ng its items in

such a way that the item run out times coincide with the run out time of

the family. A direct:.consequance is that all Items of a family will trigger

simultaneously, minimizing remnant stock , the remaining inventory held In

the items in that family .

7.2 A heuristic Approach

Max and Meal [1LI I proposed a heuristic algorithm to equalize the run

out times of the items belonging to each family. The essence of this approach*Is to allocate the family n.m quantity, Y3 , so as to maximize tie expected

tine until an item in that family runs out. Any item running out requires

scheduling the entire family ~~~~ln and, therefore, should be deferred as

long as possible within the constraints of item overstock limits and the

total family produc t ion quantity determined at the family d.tsa~~ ’egation

level.

-—

-21-

An initial run quantity is determined for each item by rr eans of the

expression :

+ k K(

(AIk_SSk)

= dk l E dk ~~~~~~~~~~I kcK(j )

where is the number of units to be produced of Item k; AI~ and SS~,

are, respectively, the available inventory and safety stock of item k;

is the forecast demand for item Ic; K(j ) is the set of

indices of all the items belonging to family 3; and Y is the total anx)unt

to be allocated for all items belonging to family 3 . Y~ was determined

by the 1’~mily disa~~ ’e~~.t ion nodel.

I~bt ice that the new run out time for item k will be:

Zk + AI k~~~SS-

kk

and, by equation (5) , thIs Is equal to

— ~

+ kcK(j)

(AIk_SSk)R Yl’k E

kcK(j )

~tiIch Is constant for every item k. This equalizes the expected runout

tine for all the items in the family. ?t reover, stunning each side of

F4uation (5) over all k values gives us:

*• E Z — Y• kcK(j )~~ ’

and , therefore , g~arantees that the total anount allocated to the family ,

has been allocated anong the , items belonging to that family .

The resulting run quantities imast be tested for ne~~tivIty and

• —22—

a~~Inst the overstock limits for each Item. If the item run quantity does

rot lie betwaen these limits It is set to zero or to the overstock limit ,as appropriate . The normalizing constant ,

Y* + E (ALK_SSk )kcKCj)

E d kkcK(j )

is appropriately nodified by eliminati ng that item from the sunmat ions

and the pro cedure is repeated again for the rema ining items .

7.2 Knapsack Approach

Bltran and Hax [1] formalized the heuristic approach by formulating

the run out time equalization problem as the following strictly convexknapsack problem for each family 3 :

*3 kcK’~”

(AIk_SS

k) Zk + AI k _ SSMinlmize ½ ~ — k

E d kkcK(j ) t—l t ].

subject to: *E Z~~~- Y

kcK(j )

Zk < O S k~~ AIk

Zk ~ nax [0~~~~~~k,t -AIk +Ss

kJ

1~e first constraint of this prob].en requires consistency in theM saggre~~tion from family to items. ~~ last t~~ constraints are the

J

upper and lowar bounds for the Item rw,~ quantities. These bounds aresimilar , to those defined fc~ the knapsack family dieaggre~~tion nodel inthe previous section.

—23—I

The two terms inside the square bracket of the objective function

represent, respectively, the run out time for family j , and the run out

time for an item k belonging to family j (assuming perfect forecast).

The minimization of the square of the differences of the run out times

will make those quantities as close as possible . The term ½ in front of

the objective function is just a computational convenience .

An algorithm to solve this problem follows very closely the logic

presented in the family dIsag~ ’egat ion algorithm. ~~tails are given in

Bitran and Hax [1] and will not be presented here .

8. Computatio na l ReBulte

We conducted a series of experiments to examine the performance of

the hierarchical system under various conditions including size of fore-

cast errors , capacity availability, rrngoitude of setup costs, length of

the plarwiing horizon, and disa~~~egat Ion methodolo~ r used from product

type to family levels.

The data used for these tests were obtal~~d fran a major manufacturer of rubber

tires. The product structure characteristics and other relevant informa-

tion are given in Figure 2. . Table 1 exhibits the demand pattern for both

product types. Product type 1 had a termin al demand season (corresponding

to tie requirements of snow tires), and consisted of 2 families and 5 items.

Product type 2 had hIghly fluctuat ing demand throughout the year and• consisted of 3 famIlies and 6 Items. Families were ~~~ups of items

sharing the same nolds in the tire curing presses, and therefore, sharing a

conro n setup cost • Items, for instance , were white wall and re~~lar wall

tire s of tie same class. Familie s and items have the same cost charac—

-~~~~~~~~~~~~~

_214_

Froduct ~ype 1: P1 Product Type 2: P2

Pl

\ ~~~~~~~P2~~~~~~~

Families: /

PI

1F1

\ [1F2~

fF\ ~P2F2

\ :Fan~ ld.es

Iten~ : I]. 12 13 II 12 12 II 12 Il 12 :Items

Family setup cost = $90 Family setup cost = $120Holding cost = $.31/unit a nonth Holding cost = $.1~0/unit a rrcnthOvertime cost $9. 5/hour Overtime cost = $9. 5/hourProductivity factor = .1 hr/unit Productivity factor = .2 hrs/unltProduction lead time = 1 nonth Production lead tine = 1 rtn th

~~gular Workforce Costs and thit Production Costs are considered fixed costs.

~bta]. Regilar Workforce = 2000 hrs/month¶I~tal Overtime Workforce 1200 hrs/nonth

FIGURE 2 . P1ODtJ~T SrRUCIURE AND (YI}ER RELEVANT Il ~~~ 1ATI0N

TThE PERIOD PRDDiJCT ‘lYPE 1 PW)DUCT ‘I?F’E 2t P1 P2

1 12,736 6~~7142 7,813 2,8553 ‘ 0 14,02314 0 14,8605 0 7.13].6 0 9,6657 1,5145 17,6038 7, 895 114,2769 10,982 11,70610 15,782 15,056

4, U 16,870 8,23212 15,870 7,88013 • 9,878 10,762

‘I~~~L 99,371 120,223

TA~ .E 1.. ~~4AND PATIEI~1S OP PT~UX~T ‘WFES

-25-

teristics and the identical productivity rates of their corresponding product

types.

‘Ire e~q erInents applied various hierarchical production planning systems

under varying conditions for a full year of sirruilated plant operations .

Production decisions were made every four weeks at which time a report was

~~rerated identifying aggregate as well as detailed decisions • The rrodel

was then updated and rerun using a one year rolling planning horizon. This

process was rep eated 13 times. At the end of the simulation, the total

setup costs, inventory holding costs, overtime costs, and backorders were

listed. A surrrnary of eleven different simulation runs is provided in Table

2. The simulations were Implemented on the Computer Based Operations

Menag~ient System (~J~~) developed at M.I.T. (see Hax and Golovin [13)) .

Run 1 can be regorded as the base case: no forecast errors , a planning

horizon of one year divided into 13 periods of 4 week durations each,f t i~~~J~tt capacity (defined as 2000 hours of regular time arid 1200 hours of

overtime per period), “normal” setup costs ($90 for families belonging to

product type 1, and $120 per family belonging to product type 2) . All

the other runs v~ ’ied some characteristics of Run 1.

8.] . Difference in Performance of Fcmiily Diaaggregation Methodologie8

%~ tested the perfornence of hierarchical planning systems using four

different family disa~~ ’egat Ion methodologies : Hax and Meal , Knapsack,r Winter s, and Equalization of Run Out Times. At the Item level, we limited

ourselves to use the heuristics approach proposed by Hax and Meal for

equal ization of run out tines. A care ful anal ysis of Table 2 indIcates

that no s1~ iIf leant dIfferences of perof’xmsnce seem to exist ameng the

four tested methodologies. Hax and Meal , and ~~~psack have a sl~~~t

—2 6—

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8~~ _______i w

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~~~

H Ni U ‘01H Ni Ui (~ rr~H 0 Ui H I Nil ~~ CoO H H O\ —3

~OO Co 0

Ui Ni Ui 0 0 Ui~~i Ni 0 N.) H H 00 0 ‘O~ Ni -~~ 0

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IHI HI HI HI

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0’. Ni~~~ ’00 -4 C’4’.0-4 0 H .14A)H0 0’. Ni~~~ ’.00

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~~~~~~ 1UiH -.1 .~~~~ ~~ Ui Vi ~~ ~~~~~~~~~~ ~~ H H

~~~~~ o~~ J~~~e U i N il0N i0 UJ~~~~~~~~Co~~~ H

—27—

advantage over Winters and Equalization of Run Out Tines, but the differences

in cost do not constitute najor g~.ins. The Equalization of Run Out Tines

procedure gives the highest setup costs, as expected. It is riot

possible to infer , with this limited azriount of experimentation , whether or

riot a given disaggregation methodo1o~~ could offer some specific advantages

under certain given condit ions . Research in progress Is desigeed to cast

some light on these Issues.

8.2 SensjtivitV to Forecast Error e

Runs 1, 2, and 3 show the in~act of forecast errors on the production

planning decisions. Forecast errors are uniformly distributed in intervals

of the type (—a,+a] and are introduced in all three levels. !‘breover,

at the family and Item levels we guarantee that the demands of families in

the same product type and the dem ands of items in a family add to the demand

of the product type and family respectively . As one ~~uld have expected ,

the quality of the decisions deteriorates under increasing forecast errors.

Both cost and size of bac1~ rders increase when- forecast errors begin to

escalate . 1-bwever , the system performs reasonably well even under fore-

cast errors of up to 30%, included in Run 3. (‘fl-e 6243 unIts back—

ordered in Run 3 of the Knapsack case represent a 97% service level.) This

is an Inportant ju stification for the hierarchical approach since, obviously,

a~~ ’e~~te forecasts can be n~re accurate than detailed forecasts.

8.3 Sensitivity to Changes in Setup Costa

J The values 1~~uted to the setup costs in the base case (Run 1) were

realistic measure s of the actual setup costs incurred .in the norval manu-

facturing operatio ns . They included direct setup costs (nenpower and

-28-

nater lals), as well as opportunity costs for having the machines idle while

perform ing the changeover . We wanted to test the system t S performance under

extreme conditions which represented unusually high setup

costs. With this purpose In mind we made two different runs , Runs 14 and 5,

with the following setup cost characteristics:

Setup CostsType l Type 2

Family 1 Fainil.y 2 Family 1 Family 2 Family 3Run 14 5000 50 1400 ~40O 1000Run 5 6000 14500 1400 5000 3000

Base Case —Run 1 90 90 120 120 120

?~.tural1y, the total cost associated with Runs LI and 5 increases sigelfi—

cantly. It can be observed that Runs 1, 4, and 5 are aincst identical in

terms of inventory ~~lding costs and overtime costs, which indicate that

the overall production strategies for these runs do not change much.

This Is to be expected as the a~~’e~~te plan does not see the increase in

setup costs. This could be a limitation of this particular hierarchical

approach when applied to situations with extremely high setup costs, since,

under these condit ions, ore could have expected hider inventory accisnula-

tbn to obtain a better trade off between inventory and setup costs.

8.4 Sensitivity to Capaaity AvailabUity

Run s 6 and 7 evaluate the perfox~~nce of the system under different

capacity conditions. Run 6 decreases tie reell*r capacity to 1660 heiws

per period; Run 7 expands the regular capacity to 2500 ~~urs (as opposed

to 2000 ~~ws in the base case). As ore could see fran the results in

able 2, the system’s performance Is quite sensitive to capacity changes.(kx~ r ti~~t capacity, there is a sig~IfIcant increase in both costs and

___I -__-

— 29—

backorders; the opposite is true under loose capacity . Clearly, the syst em

can be useful in evaluating proposals for capacity expansion.

8.5 Sensitivity to Changes in P ianning Horizon Characteristics

Runs 8, 9, 10, and 11 experizient with changing the length of the

planning horizon under different conditions. Shortening the planning hori-

zon from 13 periods to 6 periods did not affect the system’s performance

under normal capacity conditions . (Compare Runs 1 and 8 , and Runs 2 and 10).

However, as one ~~uld have expected , the size of backorders began to Increase

sigeificantly when the planning horizon is shorter under tight capacity.

Run 11 deal s with an aggeegation of tire periods In the planning horizon .

!Iie length of the planni ng horizon is still a ful l year ~ut It is divided

into only six t ime period s of uneven lengths . The first four periods have

4—week duration each, the fifth period covers 12 weeks (aggeegation of three

4—week periods), and the sixth period covers 214 weeks (aggregation of six

LI-week periods). Run 11 shows a performance quite similar to the base case .

This result might indicate that this type of aggregation of the planning

horizon could be useful in many situations , since it iimproves the forecasting

accuracy In nore distant tine periods and reduces the associated computa-

tional tine, without experiencing a decline in performance. -

8.6 Degree of Suboptiinization

Althou~~ our proposed hierarchical planning system provides optizmin

solutions to the subproblems that deal with individual decisions at each

level, obviously it Is not an overall optinun procedure . As we have pointed

out , setup costs are ignore d at the a~~’egate planning level, thus Intro-.

ducing suboptlmization possibilit ies, ‘lb analy ze how serious this subopt i-

—30—

mization problem was , we developed a mixed integer progranining (r ’tEP) nodel

at a detailed item level to identif y the true optima.l solution to our test

problem. The ~~P nodel was implemented by means of IBM ’s £~ SX/7~’aP code ,

which is a general purpose branch and bound algerithm.

IXe to the computational cost of solving ~~P rro dels, we limited

our c~iparison s between the hierarchical planning system and the ~ttP nodel

to those situations containing no fore castin g errors • In those cases ,

we could solve the MIP nodel only once , and obta in the optirrn.u n yearly

cost . (If forecast errors would have been introduced we would have had

to solve the MIP nodel 13 t ines for each run , which was prohibitively

expensive.)

We cor~uted MIP solutions to three of our previous runs : the base

case (Run 1’), the first high setup cost run (Run L I ’ ) , and the tight capa-

city run (Run 6 ’) . The MEP results are given In Thble 3. The existing

limits on the node tables of the branch and bound code used did not allow

us to determine the true optlsnz.un in the MIP runs. Therefore, the solutions

reported in ‘l~ble 3 might still be improv ed. ~~ble 3 also provides the

contIrnx~us lower bounds obtained at the time in which the computations

1’ 4’ 6’RUN HI~~~Setup Cost

BASE CASE CASE I ~~~ T CAPACITYO~ T ~) FO~~CAST ERRJR P1: 5000 50 1600 Reg. Mrs.

______________________ P2:_ 400 ,400 ,10000

_________________

~~‘lUP ~l ,590 48 ,050 3,930IUDIl~ 75,953 79,880 115,872OVERrIr€ 77~796 75,430 117,430

‘IVTAL COST(Best 1Q~own 158 ,339 203,360 237,232solution)

LOWER BOL’~D 153,926 162,783 233,665

TARE 3. SUt.T4P~~ OF 001.WTJFATIONAL ~ SIJLTS WITh ~ff)~D ThTFX]ER P~~~ A*T~JG ~‘O~~IS

-31-

were Interrupted . For all practica l purpo ses, we could consider the solu-

tions corresponding to Runs 1’ ar id 6’ to be optimal . Possibly Run 14’

could still be improved.

By comparing the total costs of the three runs for the Knapsack case

we have :

Knapsack Best knownHierarchical System MtP solution

Pase Case 158 ,981 158 ,339High Setup Cost 220 ,535 203,360Tight Capacity 236,733 237, 232

We see that the hierarchical planning syst em was extremely efficient . It

appears that only under abnormally high setup cost the system’s perofnnance

begins to depart si&ilficantly from the overall optin~]. solution.

In stmnary , the hierarchical systems seems to perform near optimi.zn

when setup costs are noderate. ~~~ base case cost, which reflected the

operating data of the tire industry, is only • 14 percent hider than the

best known optim~in solution obtained by the MEP formulation. However, the

coat of each run of the hierarchical system was approxinately $5, while the

corresporriing MIP run cost rear $50. In addition, the MIP approach would be

cai~utat iona lly Impossible to carry out fbr larger problems. rbreover, the

hierarchical system appea rs to offer coherent solutions wxler vary ing fore-

cast errors, capacity availabilities , and planning horizon lengths .Extr ~~~ly high setup costs could affect the performence of the system.

In practice , familie s with very high setups are candidates for contlnt~ usprod uction (as opposed to batch production) I! they have a high level ofdeii~nd. In such a case, those families can be handled independently ofthe hierarchical system. In situations where there axe few high setupfamilies with low demaj~xt , special constraints can be imposed on the family

—32—

disa~~ e~~t ion node 1 to produce those families in large enough quantities .

This can be accomplished by setting the lower bound of the family to Its

unconstrained economic order quantity. When all the f’ainilles in the

product structure have high setup costs arid low demand levels, it might

not be desirable to eliminate setup costs at the aggregate level. In

that situation , we could eliminate the ag~ ’egat e planning nodal for por’duct

types arid allocate prod uction quantities at the family level by using

an approach similar to that prop osed by Lasdon and Terjtng [16]. We would

then apply the item disag~ ’egation nodel to allocate the family production

quantities anong items.

References

1. Bitrañ, 0. R. and A. C. Hax; “On the L~sIgri of Hierarchical ProductionPlanning Systems”, 1~ cision Sciences 8 (1), 28—55 (1977).

2. Bitran, 0. R., and A. C. Flax; “On the Solution of Convex KnapsackProblems with Bounded Variable s” , Massachusetts Institute of Technolo~ r ,Operations Research Center , Technical Repor t No. 129, April 1977 .

3. Bradle y, S. P. , A. C. Flax , and T. L. Mageanti; Applie4 MathematicalPr ograninTh ,~~ Addison Wesley, 1977. Chapter 6 based on the technicalpaper “Inte~~’ation of Strateg ic and Tactical Planning in the A1~.zniri~nIndustry” by A. C. Flax, Operations Researc h Center , M .I.T . WorkingPaper 026—73, September 1973; Chapter 10 based on the paper by~~bert J. Armstrong and Arno].do C. Flax, “A Hierarchical Approachfor a Naval Tender Job-Shop t~si~~”, Operations Research Center ,M.I.T., Technical Report Ho. 101, Au~~st 1974.

4. Bro,m, R. 0.; !~cIsion Rules for Inventory Mana~ men,~ Holt, Rinehart,and Winston, 1967.

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